Chapter 10

TEMPORAL LOGIC

Yde Venema

1 Introduction

Time must be the most paradoxical concept our minds have to deal with.To quote from the Confessions of St. Augustine

What then, is time? If no one asks me, I know; but if I wish toexplain it to someone who should ask me, I do not know.

One of time’s most puzzling aspects concerns its ontological status: on theone hand it is a subjective and relative notion, based on our conscious ex-perience of successive events; yet on the other hand, our civilization andtechnology are based on the understanding that something like objective,absolute Time exists. Some philosophers have taken this paradox so far asto conclude that time is unreal; others, accepting the existence of absolutetime, have engaged in heated debates regarding its structure, be it linear orcircular, bounded or unbounded, dense or discrete.

But even if we leave these metaphysical issues aside, it is obvious thattime plays such a fundamental role in our thinking that there is a clear needfor precise reasoning about it, such as we see in Physics, formal Linguistics,Computer Science, and Artificial Intelligence. While these enterprises arenot necessarily concerned with the same concept of time, they all could gounder the heading of Temporal Logic. Often, however, a more restricted,technical definition is used in which temporal logic—or tense logic—is abranch of modal logic, an approach that began about forty years ago withthe work of Arthur Prior. We will largely confine ourselves to this modalperspective here, though, as we shall see, this still includes a great varietyof systems.

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In the next section we discuss some of the most well-known mathemat-ical modellings of time. These are the structures the formal languages oftemporal logic are designed to talk about. The main part of this paper, Sec-tion 3, is devoted to a fairly detailed exposition of Prior’s basic tense logic;the aim of this is not only to present this particular system, but perhapseven more to introduce the kinds of questions that temporal logicians tendto ask. In the Sections 4 and 5 we describe some extensions and alternativesto this base system. In Section 6 we sketch some developments that havetaken place over the last ten years or so. Finally, in the Epilogue we try toanswer the question what Temporal Logic is.

2 Flows of Time

Before we can start a discussion of various logics of time, it helps to look atsome standard mathematical models of time. When asked to think of timein an abstract way, many people will form a picture of a line — only the sim-plest of the many spatial metaphors that people use for temporal concepts!The mathematics of this picture is given by a set of time points, togetherwith an ordering relation and perhaps a metric measuring the distance be-tween two points. Later on, we will discuss some objections and alternativesto this point-based paradigm. For now, let us formally represent time as aframe; that is, a structure T = (T,<) such that < is a binary relation onT , called the precedence relation. Elements of T are called time points; ifa pair (s, t) belongs to < we say that s is earlier than t. In the remainderof this section we will discuss a number of more or less intuitive conditionsthat have been imposed on such structures in order to make them usefulas models of time. (We will frequently use first and second order logic fordescribing these properties; the first order frame language we use will haveonly one dyadic predicate symbol, which is denoted by R and interpreted as<.)

Obviously, many frames will not qualify as intuitively acceptable repre-sentations of time. At a minimum one should require that < be irreflexiveand transitive. A frame satisfying these conditions will be called a flow oftime. Flows of time are known from mathematics as strict partial orders,and in accordance with this we will use familiar notation like s > t for ‘sis later than t’ or s ≤ t for ‘either s = t or s < t’. For a point t, the set{s ∈ T |t < s} will be called the future of t; the past of t is defined likewise.(In the sequel, we will omit definitions pertaining to the past if they mirroran obvious counterpart for the future.)

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Standard candidates are given by the familiar orderings of well-knownnumber sets: N = (N, <) (the natural numbers), Z = (Z, <) (the integernumbers), Q = (Q, <) (the rational numbers) and R = (R, <) (the realnumbers). Less familiar examples are the binary tree B = (B,≺) (whereB is the set of sequences of 0s and 1s, while s ≺ t holds if s is an initialsegment of t), and four-dimensional Minkowski spacetime S = (R4,�); herewe put (x0, x1, x2, t) � (x′0, x

′1, x′2, t′) if not only the temporal component of

the first point is smaller than that of the second one (t < t′), but also thespatial distance between the two points should enable one to reach the onepoint from the other without having to travel faster than the speed of light.

Observe that our definition excludes circular time: if there were a series oftime points s1 < s2 < . . . sn < s1 then by transitivity we would have s1 < s1

which is not possible since we assumed our flow of time to be irreflexive.Since it is not the logician’s task to choose between different ontologies, whynot allow circular time? After all, many civilizations have regarded timeas being essentially cyclic in nature. Also, practical applications of circulartime are easily conceivable, such as the construction of rotas. Our onlyreason is simply that circular time has received very little attention in thelogical literature.

On the other hand, the reader may have missed one condition in thedefinition of a flow of time, namely linearity. A strict partial ordering iscalled linear if any two distinct points are related; expressed in first orderlogic, the structure is to satisfy the sentence ∀xy (Rxy ∨x = y ∨Ryx). Thisperspective on time is dominant in science and, probably for that reason, hasbecome the standard in most people’s minds; in particular, all of our numberexamples are linear flows of time. Nevertheless, so-called branching-timestructures such as B and S have received a lot of attention in the literatureon temporal logic. A structure is called branching to the future if there issome point having two unrelated points in its future, and not branching tothe future, if on the contrary, the future of each point is linearly ordered.A flow of time is not branching if it is neither branching to the future norbranching to the past; note that this condition differs from linearity in that itdoes not exclude ‘parallel’ time lines. In the literature one often encountersthe condition that flows of time are allowed to branch to the future, butnot to the past; this condition reflects the idea that at any moment, thepast is determined while the future is not. As we will see later, the logicof branching time ties up with the logic of necessity and possibility, that is,with alethic modal logic [See Chapter 7]. In the sequel of this section wewill confine ourselves to linear time, but this is not to say that the conceptsto be defined would not make sense outside of this context.

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Questions concerning the boundedness of time have occupied philoso-phers, theologians and physicists for centuries, but for the logician this isgenerally not the most interesting issue. Let us just mention the definitionspertaining to the future: T has a first point if it satisfies ∃x∀y (Rxy∨x = y),while it is called right-serial if each point has a non-empty future.

A more fundamental choice, it seems, is that between denseness anddiscreteness of a flow of time. A linear ordering T is dense if betweenany two distinct points we can find a third point; formally: ∀xy (Rxy →∃z (Rxz ∧ Rzy)). Because this way of representing time is very convenientfor modeling the notion of movement, we have become quite familiar withdense flows of time such as the orderings of the rational or the real numbers.However, for computer scientists and economists time has a very differentflavor in that it is supposed to proceed in discrete steps; that is, with eachnon-final point we associate a next point or immediate successor. In firstorder logic: ∀xy (Rxy → ∃z (Rxz ∧ ¬∃u (Rxu ∧Ruz))). Standard examplesof discrete flows of time are given by the natural or the integer numbers.

Density is often confused with continuity. Suppose that we cut theset of rational numbers into a left and a right half, of numbers smaller andbigger than

√2, respectively. Such a cut, without a proper point on either

edge, is called a gap, and a flow of time is called continuous if it has nogaps. Q thus forms the standard counterexample, whereas R and Z arecontinuous.

Unlike the properties discussed before, continuity is essentially a secondorder notion, its definition necessarily involving a quantification over setsof time points. There are many other interesting second order conditionsthat one may impose on temporal structures. For example, one might arguethat abstract time exists independently of the events ‘filling it’, and thattherefore, the structure of time should be ‘the same everywhere’. One wayof making this precise is to demand that a flow of time is homogeneous: forany two points s and t of T , there should be an automorphism of T (thatis, a bijection f from T onto T satisfying x < y iff f(x) < f(y) for all xand y in T ) mapping s to t. Another second order property that we willmeet further on is that of having finite intervals; this means that there canbe at most finitely many points between any two points. Observe that thiscondition implies discreteness of both < and >.

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3 Basic Tense Logic

In this section we will show temporal logic at work. That is, we introducePrior’s basic system of temporal logic, and discuss some of the fundamentallogical questions pertaining to it.

Syntax and Semantics

In order to define the syntax and semantics of temporal logic, we shouldfirst note that temporal logic is an extension of classical propositional logic.Recall that classically, propositional formulas are interpreted as truth values(either 1 for ‘true’ or 0 for ‘false’); this truth value is inductively determinedby a valuation: a function mapping propositional variables to truth values.Once we know the valuation, the truth value of any formula is fixed. Nowwhat to do with the fact that the truth value of statements like ‘it is raining’or ‘I am wearing an umbrella’ will change from time to time? For instance,it may be raining today but sunny tomorrow; or, I may be wearing myumbrella now but fold it in some time after the rain stops.

The first basic idea underlying temporal logic is to address this issue bymaking valuations time-dependent; more precisely, one associates a separatevaluation with each point of a given flow of time. Formally, let T = (T,<)be a flow of time; a valuation on T is a map π : (T → (Φ → {0, 1})); hereΦ denotes the set of propositional variables. A model is a pair M = (T , π)consisting of a flow of time and a valuation.

Observe that with this definition we can already interpret classical for-mulas in each point of a model, in a standard way. For instance, we will saythat the formula p∧¬q is true at a time point t precisely if π(t)(p) = 1 andπ(t)(q) = 0. The spice of temporal logic, however, lies in its second basicidea, namely to use new, non-classical connectives to relate the truth of for-mulas in possibly distinct time points. In this section, we discuss two suchoperators: F and P . These names are mnemonics for ‘future’ and ‘past’respectively: the intended meaning of the formula ‘Fϕ’ is ‘at some time inthe future, ϕ is the case’, while ‘Pϕ’ is to be read as ‘at some time in thepast, ϕ holds’.

Formally, we define the set Lt of Priorean formulas as the smallest setcontaining the propositional variables that is closed under constructing newformulas using the boolean connectives ¬ and ∧, and the temporal operatorsG and H. For technical reasons, we take F and P to be defined operatorsin our set-up; Fϕ abbreviates ¬G¬ϕ and Pϕ abbreviates ¬H¬ϕ. Gϕ andHϕ are read as ‘henceforth, ϕ’ and ‘hitherto, ϕ’, respectively. As further

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abbreviations we use ⊥, >, ∧, → and ↔ in their usual meaning. We willalso frequently refer to the mirror image of a formula; this is simply theformula one obtains by simultaneously replacing all Hs with Gs and viceversa.

Bringing the previous observations together, we can give the followinginductive definition of the notion of truth of a formula ϕ at a time point tin a model M = (T,<, π):

M, t q if π(t)(q) = 1,M, t ¬ϕ if not M, t ϕ,M, t ϕ ∧ ψ if M, t ϕ and M, t ψ,M, t Gϕ if M, s ϕ for all s with t < s,M, t Hϕ if M, s ϕ for all s with t > s.

(1)

If M, t ϕ we say that ϕ holds or is true at t.As an example, consider the ordering N of the natural numbers; let τ be

the valuation making q true at all numbers bigger than 1 000, and r at alleven numbers. With this valuation, it is easy to see that the formula FGqholds at the point 0. For, the formula Gq holds at those points of whichthe future is a subset of the set of ‘q-points’, and this is the case for anynumber bigger than 999. But from M, 1 000 Gq and 0 < 1 000 it followsthat M, 0 FGq. It is likewise easy to see that the formula FGr does nothold at 0, or indeed, at any point in this model; the formula GFr on theother hand holds throughout M.

Finally, observe that from the technical point of view, this system is verysimilar to the systems defined in Chapter 7 on Modal Logic: G and H arevery much like the operator L of alethic modal logic. The difference is thatin Priorean temporal logic we are dealing with two modal operators insteadof one. One might then expect that we would interpret this language instructures with two accessibility relations, say, RF and RP . And in factwe may adopt a perspective in which we see < and > as these two distinctaccessibility relations; however, it is a crucial aspect of temporal logic thatthese two accessibility relations are each other’s converse. The main dis-tinction between alethic modal logic and temporal logic is thus one of aim:temporal logic starts with structures (flows of time), for which one is tryingto find good modal description languages; whereas in alethic modal logic ithas more often been the other way around.

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Validity and Definability

Temporal logicians are generally not so much interested in the truth or falsityof formulas in specific models, but rather in those formulas that remain truethroughout the flow of time even if we change the valuation. It is felt thatsuch formulas provide essential information concerning the structure of theunderlying flow of time. Formally, we say that a formula ϕ is valid on a flowof time T , notation: T ϕ, if for every valuation π on T , and every pointof T , we have (T , π), t ϕ. A formula is valid in a class of flows of time if itis valid on each member of the class. The notion of satisfiability is defineddually: we say that a formula ϕ is satisfiable in a flow of time (a class offlows of time) if its negation is not valid on the flow of time (in the class offlows of time, respectively).

As an example, we show that the formula Fq → FFq is valid on the classof dense linear orderings. Assume that T is a dense linear flow of time; inorder to show that Fq → FFq holds on it, consider an arbitrary valuation πon T , and an arbitrary point t in T such that (T , π), t Fq. By the truthdefinition, there is a later point s where q holds. But by density, there mustbe some point u between t and s; from u < s we derive that Fq holds at u;but then from t < u we may infer that FFq holds at t; since t and π werearbitrary, this suffices to show that T Fq → FFq.

On the other hand, it is easy to see that the formula Fq → FFq isnot valid on the ordering of the integers. For, take the points 0 and 1 andconsider the valuation π that makes q true only at 1; then obviously, Fq istrue at 0; but since there is no integer number between 0 and 1, the formulaFFq cannot be true at 0. This shows that indeed Z 6 Fq → FFq. We canin fact generalize this argument to show that the formula Fq → FFq can befalsified on every non-dense frame. For, any non-dense frame must containtwo points s < t without intermediate points; so the valuation making q trueonly at t will make the formula Fq → FFq false at s. Hence, the formulaFq → FFq is very informative; it is a reliable witness of the density of aflow of time.

In general, we say that a Priorean formula ϕ defines a class C of flows oftime within a class K if for every flow of time T in K, T ϕ iff T belongsto C. If C is given as the class of frames satisfying some first order propertyα, we also say that ϕ corresponds to α (within K). For instance, we havejust seen that the formula Fq → FFq corresponds to density.

Not every property of flows of time is definable; for instance, we canprove that there is no Priorean formula that defines the class of branchingflows of time. On the other hand, there is a formula defining the flows of time

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that are not branching; for, the formula PFq → (Pq ∨ q ∨ Fq) correspondsto non-branchingness to the future. Hence, the conjunction of this formulaand its mirror image defines the flows of time that are not branching.

Especially if we confine ourselves to linear orderings, many interestingproperties of flows of time can be defined in the Priorean language. Thefollowing table lists a number of such correspondences holding for linearflows of time; here 3ϕ abbreviates Pϕ ∨ ϕ ∨ Fϕ, and 2ϕ ≡ Hϕ ∧ ϕ ∧Gϕ.For future reference, we have given names to the modal formulas.

having a first point H⊥ ∨ PH⊥ (A1)left-seriality P> (A2)having a final point G⊥ ∨ FG⊥ (A3)right-seriality F> (A4)discreteness (F> ∧ q ∧Hq)→ FHq (A5)density Fq → FFq (A6)continuity (Fq ∧3¬q ∧2(q → Hq))→

→ 3((q ∧G¬q) ∨ (¬q ∧Hq)) (A7)having finite intervals G(Gq → q)→ (FGq → Gq) ∧

∧ H(Hq → q)→ (PHq → Hq) (A8)

Finally, since Priorean formulas may be interpreted on all frames (alsoones that are not strictly partial orders), the question naturally arises whetherthe class of flows of time itself is definable. Since, analogous to the case of or-dinary modal logic, transitivity may be defined by the formula Gp→ GGp,this boils down to the problem of finding a correspondent for irreflexivity(within the class of transitive frames). Unfortunately, there is no such for-mula.

Axiomatics

As we mentioned already, temporal logic starts with flows of time; but ob-viously, this does not diminish the interest in finding complete calculi forvarious classes of flows of time. Obviously, there are close connections withthe axiomatics of alethic modal logic as discussed in Chapter 7. In particu-lar, analogous to K, there is a minimal tense logic for the Priorean languageas well; it is called Kt and defined as the smallest class of Priorean formulasthat is closed under the following axioms and derivation rules:

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(CT ) all classical propositional tautologies(DB ) G(q → r)→ (Gq → Gr)

H(q → r)→ (Hq → Hr) (Distribution)(CV ) q → GPq

q → HFq (Converse)(4 ) Gq → GGq (Transitivity)(US ) if ϕ is a theorem, then so is ϕ[ψ/q] (Uniform Substitution)(MP) if ϕ and ϕ→ ψ are theorems, then so is ψ (Modus Ponens)(TG) if ϕ is a theorem, then so are Gϕ and Hϕ (Temporal Generalization)

Here ϕ[ψ/q] denotes the result of substituting the formula ψ for the propo-sitional variable q, uniformly throughout ϕ.

Most of these axioms and all of these rules are, perhaps under differentnames, familiar from ordinary modal logic. The exception is the Converseaxiom (CV); as we will see, this axiom is needed to ensure that the accessi-bility relations for the operators G and H are each other’s converse. In thechapter on Modal Logic it is discussed in detail that the formula (4) reflectsthe transitivity of the intended accessibility relation of a modal operator;thus, our constraints on flows of time explain the presence of (4) as an ax-iom. Recall that in the previous section we already saw that the property ofbeing irreflexive is not definable in the Priorean language; now we see thatirreflexivity does not even yield any extra validities. (This is not the rule inmodal logics: frame conditions that are not definable in the modal languagemay nevertheless imply the validity of modal formulas.)

Theorem 3.1 The logic Kt is sound and complete with respect to the classof all flows of time.

For lack of space, we omit the proof of Theorem 3.1. Instead, we con-centrate on completeness for the class of linear flows of time. Let Lin bethe extension of Kt with the axiom (NB), which is the conjunction of theaxiom PFq → (Pq ∨ q ∨ Fq) (defining non-branching to the future) and itsmirror image FPq → (Fq ∨ q ∨ Pq).

Theorem 3.2 The logic Lin is sound and complete with respect to the classof linear flows of time.

Proof. Just as the completeness proofs of Chapter 7, our proof methodwill make use of canonical models. Hence, let W c be the set of maximalLin-consistent sets of formulas (for unexplained terminology we refer to themodal completeness proof), and define the relation Rc on W c by Rcwv iff

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ϕ ∈ v for all Gϕ ∈ w. The structure F = (W c, Rc) is called the canonicalframe; on it, we define the canonical valuation πc so that πc(q)(w) = 1 iffp ∈ w.

Our first aim is to prove a Truth Lemma for this model, stating thatfor all Priorean formulas and every point w of the canonical model Mc =(Fc, πc) we have that ‘truth coincides with membership’:

Mc, w ϕ iff ϕ ∈ w.(2)

Analogous to the proof of Theorem 4 in Chapter 7, (2) is proved by formulainduction. There is only one minor problem, caused by the fact that wenow have two modal operators, and only one accessibility relation. This isprecisely where the Converse axioms come in: they enable us to show thatthe canonical accessibility relation does not only work well for G but alsofor H. For, we can prove (details are left to the reader) that Rcwv iff ϕ ∈ wfor all Hϕ ∈ v.

Now it follows easily from (2) that every Lin-consistent set of formulasis satisfiable in the canonical model, but unlike the case of modal logicslike S4 we are not finished here. We need to satisfy our Lin-consistentset of formulas in a linear flow of time. Now it is easy to verify that thecanonical accessibility relation is transitive (use the axiom (4), as in themodal completeness proofs); it is not very difficult to show that Rc is notbranching (but we leave the details of this proof to the reader — use theaxiom (NB)); but it is impossible to prove that Rc is a linear ordering,because in general this will not be true! The main problem is that nothingguarantees irreflexivity of canonical accessibility relation. The difficult partof the proof consists in showing that we can transform the canonical frameinto a strict linear order, while truth of formulas is preserved.

Let us agree to call a frame F = (W,R) a pseudo-line if R is transitiveand strongly connected (that is, satisfying ∀xy(Rxy ∨ x = y ∨ Ryx)). Nowgiven any maximal Lin-consistent set Σ, we may restrict ourselves to thepart of the canonical frame that is connected (via Rc) to Σ and still provethe analogue of the Truth Lemma (2). It thus follows that every consistentformula is satisfiable in a pseudo-line. But then the missing link in the proofof the completeness theorem for Lin is the following claim.

If ϕ is satisfiable on a pseudo-line, then also on a linear flow of time.(3)

In order to prove claims like (3), several methods of ‘frame surgery’ havebeen developed; in order to give the reader an idea of such techniques, webriefly sketch the bulldozing method here. Assume that ϕ is satisfiable in

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the model M = (F , π) based on the pseudo-line F = (W,R). The firstobservation is that F may be represented as a linear ordering ≺ of so-calledclusters which are special subsets of W . Each point s of W belongs to aunique cluster Cs which is either degenerate (consisting of a single irreflexivepoint) or proper (if R is universal on it). The relation ≺ is defined such thatCs ≺ Ct if and only if Cs 6= Ct and Rst.

The key idea is now to ‘bulldoze’ each proper cluster into a special linearordering LC and to replace each C with LC . Obviously, replacing eachproper cluster with a linearly ordered model yields a linear order; but is ϕstill satisfiable in the new model? To understand the positive answer to thisquestion, note that any proper cluster introduces a infinity of informationrecurrence in both the forward and backward directions: we can follow pathswithin C, moving either forwards and backwards along R, for as long as weplease. Thus, when we replace a cluster C with a linear ordering, we mustensure that the linear ordering duplicates all the information in C infinitelyoften, and in both directions. Bulldozing does precisely this, in the moststraightforward way possible. For instance, suppose that the cluster C hasthree elements only: s0, s1 and s2, with associated classical valuations σ0,σ1 and σ2. Then LC is given as the model (Z, πC); here πC is given byπC(z) = σz mod 3; that is, LC consists of an unbounded (in both directions)series of points with associated classical valuations σ0, σ1 and σ2; as in· · ·σ1σ2σ0σ1σ2σ0 · · ·.

There is thus an obvious relation linking points in the new, transformedmodel to points in the old one; using this we may prove that ϕ is indeedsatisfiable in the new model. This finishes the proof sketch of (3). qed

Turning to the axiomatics of specific structures, let us define the followinglogics:

Lin.N: Lin +A1 +A4 +A8Lin.Z: Lin +A2 +A4 +A8Lin.Q: Lin +A2 +A4 +A6Lin.R: Lin +A2 +A4 +A6 +A7

For these logics we have the following result.

Theorem 3.3 The logics Lin.N, Lin.Z, Lin.Q and Lin.R are sound andcomplete axiomatizations of the set of validities of the flows of time N , Z,Q and R, respectively.

We may conclude that temporal logicians have been rather successful inaxiomatizing the standard flows of times and the most natural classes of

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flows of time. Nevertheless, it would be wrong to conclude that conversely,(axiomatically defined) tense logics are always characterized by a class offlows of time. As in modal logic, incompleteness is the rule; in fact, the veryfirst example of an incomplete (poly-)modal logic was found in tense logic.

Decidability and Complexity

The completeness theorems that we mentioned in the previous subsectionare all very nice, but of course, if one wants to do actual reasoning in one ofthese logics, further properties are required. Minimally, one wants the logicto be decidable; that is, the existence is required of a terminating algorithmseparating the logic’s theorems from its non-theorems. Fortunately, all thecomplete logics defined in the previous subsection have this property. Wemention only the following results explicitly.

Theorem 3.4 The Priorean tense logics of the classes of all flows of time,and of all linear flows of time, are decidable.

This follows from the fact that these logics are finitely axiomatizableand have the finite model property. The latter may be proved through themethod of filtrations or the method of minimal models of Chapter 7, withallowance for complexities analogous to the proof of completeness for Lin.

For practical purposes decidability is not enough, however; one would liketo have an efficient calculus. A more fine-grained analysis is needed to revealthe computational complexity of temporal logics. There is not enough spaceto go into details here; we only mention the result that the satisfiabilityproblem for linear time is in NP. To be more precise, one can devise anon-deterministic Turing machine algorithm that correctly tells whether aPriorean formula ϕ is satisfiable in a linear frame or not, while coming upwith this answer within f(ϕ) computation steps. Here f is a linear functionthat grows at the same rate as the length of the formula ϕ.

4 Extending the Language

Since and Until

Our discussion in the previous section was based on the basic temporallanguage having G and H as its only primitive operators. For many ap-plications, however, this language is too poor in expressivity, and severalextensions with new operators have been suggested. The most importantof these are the dyadic operators S and U introduced by H. Kamp; their

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intended meaning is respectively ‘since’ and ‘until’, as in the sentences ‘Eversince the roof caved in, it’s been wet in the house’ and ‘Until we get the rooffixed, it will be damp in the house’. Let Lsu denote the extension of the Pri-orean language with these two new connectives, the formal truth definitionof which is given as follows.

M, t Uϕψ if M, s ϕ for some s such that t < sand M, u ψ for all u with t < u < s,

M, t Sϕψ if M, s ϕ for some s such that s < tand M, u ψ for all u with s < u < t.

It is interesting to observe that the ‘old’ operators can be expressed inthis new language, for instance Fϕ may be seen to abbreviate Uϕ>. Butconversely, the new operators really add expressive power to the language;we can prove that they cannot be defined in terms of the old.

Another interesting temporal operator is the so-called nexttime or to-morrow operator X; the formula Xϕ holds at a time point t if ϕ holds atthe next moment in time (if there is such a next moment). Obviously, suchan operator only makes sense in a discrete flow of time, as, for instance, incomputer science, where one wants to talk about the next state of a process.However, adding this new connective to Lsu would not add any expressivepower, since Xϕ can already be defined as an abbreviation for Uϕ⊥.

This raises the question whether perhaps every temporal operator can bedefined in this apparently expressive language Lsu. The answer to this ques-tion is positive; that is, we can prove some sort of functional completenessresult for Lsu.

Theorem 4.1 (Kamp) Over the class of linear, continuous orderings, ev-ery temporal operator can be defined in Lsu.

where by ‘temporal operator’ we mean any operator whose truth definitionis expressible in first order logic. The restriction in the theorem to certainflows of time is essential. In particular, once we drop the condition oflinearity, the results tend to be negative; for instance, over the class of allflows of time we cannot find a finite expressively complete set of operators.

Finally, for the language Lsu one can ask the same kind of questions asfor Lt; and indeed, several results have been proved concerning definability,axiomatizability and decidability. In general these results are positive, butfor lack of space we cannot go into detail here.

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Branching Time Languages

As mentioned above, allowing flows of time that branch to the future meansthat we can no longer assume that the past determines everything that isgoing to happen. But if our formalism has to take into account that thereare many different courses of events possible, it seems appropriate to paysomewhat more attention to the truth definition of our future operator F .For, the intuitive meaning of Fϕ, namely ‘it will be the case that ϕ’, is nowmore ambiguous than in linear flows of time. Recall that the interpretationof Fϕ that we may calculate from the truth definition (1) yields ‘ϕ holds atsome future moment of some possible course of events’. But it does not seemto be unreasonable to assume that ‘it will be the case that ϕ’ expresses thespeaker’s conviction that ϕ will be the case, in the actual course of events,or perhaps no matter what course of events. These two interpretations giverise to respectively the Ockhamist and Peircean schools in branching timelogic.

In order to compare these two approaches, assume our flows of time tobe trees, that is, connected strict partial orders that do not branch to thepast. (Connectedness forbids for instance parallel time lines.) A branch ofa tree T = (T,<) is a maximal linearly ordered subset of T ; the intuitiveidea is that each branch through t represents a possible course of events (fora point t and a branch b, we say that t lies on b or that b goes through t ift belongs to b). In this way, we can imagine a possible future of t as the setof all later points on some fixed branch b through t; since T is a tree, eachpoint will have a unique past.

Now Peircean branching time logic interprets the proposition ‘it willbe the case that ϕ’ in the second way indicated above, namely that ϕ isbound to happen in every possible future. To make this more precise, definethe Peircean tense language as the extension of the Priorean one with thefuture operator F2; this operator has a second order definition, involving aquantification over all branches through the actual time point:

M, t F2ϕ if on each branch through t there is some s > t with M, s ϕ.(4)

In the Ockhamist approach on the other hand, it is meaningless to askabout the truth value of formulas of the form Fϕ or Gϕ at a time pointt, unless we have specified which of the possible futures of t we have inmind. In order to be able to express that something that will be the case nomatter what form the future will take, Ockhamists extend the language withan alethic modal operator 2. Ockhamist tense logic is thus an interestingcombination of modal and tense logic; perhaps the easiest way to work out

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the idea formally, is to require that in Ockhamist semantics the truth valueof any formula is evaluated at a pair consisting of a time point and a branchthrough this point (representing the actual course of events). We thus arriveat the following truth definition.

M, t, b q if π(t)(q) = 1,M, t, b ¬ϕ if not M, t, b ϕM, t, b ϕ ∧ ψ if M, t, b ϕ and M, t, b ψ,M, t, b Gϕ if M, s, b ϕ for all s on b with t < s,M, t, b Hϕ if M, s, b ϕ for all s on b with t > s,M, t, b 2ϕ if M, t, c ϕ for all branches c through t.

(5)

It is interesting to note that the Peircean language can be seen as afragment of the Ockhamist one; consider the inductively defined translation(·)o mapping Peircean formulas to Ockhamist ones. The only non-trivialclause of this map concerns the future operators: (F2ϕ)o = 2Fϕo and(Gϕ)o = 2Gϕo. It is straightforward to prove that for all tree models M,all points t in M and all branches b through t, we have that

M, t ϕ iff M, t, b ϕo.

Many results are known concerning Peircean and Ockhamist logic; forinstance, axiomatizations have been found for the Peircean logic of the classof all trees. This logic is also known to be decidable, as is its Ockhamistalternative. It is an outstanding open problem to find an explicit axiomati-zation for the Ockhamist tree logic.

Finally, it is obvious that one can extend these branching time logics evenfurther, for instance with the Since and Until operators defined earlier. The‘future fragment’ of such systems is closely related to so-called computationaltree logics that have been developed within theoretical computer science forthe purpose of reasoning about paths through labeled transition systems,which in their turn form perhaps the simplest mathematical models of thenotion of computation. It is interesting to note that the Peircean and theOckhamist approaches in philosophical logic find (much more technicallyinspired) counterparts in the development of the computational tree logics:CTL and CTL∗, respectively.

5 Time Periods

So far we have applied a point-based paradigm to represent time. Neverthe-less, it seems that in every field where temporal logics are used or studied,

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at a certain moment systems are designed in which periods are the centralentities, or at least, play a more prominent role.

Motivations The point-based perspective has never been without philo-sophical objections. For instance, Zeno’s paradox of the flying arrow, which,it is argued, cannot change position at a isolated moment of time and thuscannot move at all, makes it clear that there is something problematic con-cerning the representation of time as a series of durationless moments ifwe want to describe the concept of movement. Some temporal predicatesseem simply not to apply to time points. Suppose that p is a propositionformalizing the statement that Zeno’s arrow moves. Obviously, the flyingof an arrow is an activity that is extended in time; hence, one might arguethat it is pointless to evaluate the truth of p at moments of time. It thusseems that we at least need the existence of time periods for the evaluationof certain expressions.

Apart from such semantic considerations, it is clear that time points arenot the kind of objects that we can directly perceive. Due to years of expo-sure to the scientific view on time we may not always realize this, but if wewant to base reality on our direct experience, then time points will come outas highly abstract and complex artifacts. Thus, it has been argued, it is adubious enterprise to take points as having primitive ontological status; peri-ods form a far sounder base. This second argument has been taken up, witha more practical twist, within Artificial Intelligence. Here the idea has beenadvocated that period-based representations of time are simpler and morenatural in formalizing common sense reasoning than the standard scientificmodels. (Obviously, this argument may be pushed further, questioning theNewtonian perspective in which absolute Time exists regardless of anythinghappening in it. Such objections may lead to event-based ontologies whichdue to lack of space we cannot discuss here.)

Finally, in our discussion until now we have assumed that there is a clearand intuitive distinction between points and periods. This is questionableas well, however; one can quite convincingly argue that there is a notionof granularity involved here. A good example can be taken from ComputerScience, where the addition of two numbers may be taken as an atomic,durationless action of a high-level programming language, whereas it may beimplemented in terms of many operations on the lower level of the machinelanguage.

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Time in periods It is important to observe that the need for a moreprominent role of periods does not necessarily commit one to model timein structures in which periods are primitive entities; they might as well bederived objects.

Indeed, one could well start from a flow of time T = (T,<) as describedearlier, and then consider the question how to represent chunks of timewithin such a structure. For instance, periods could be defined as convexsets: subsets C of T that are uninterrupted in the sense that whenever s anda later point t belong to C, then so does any point between s and t. A set-theoretically slightly simpler option is to only consider (closed) intervals; inthis approach, the period {u ∈ T | s ≤ u ≤ t} can simply be represented asthe pair [s, t]. Observe that this approach has the advantage that propertiesof periods can be expressed by binary predicates in the first order framelanguage, whereas for convex sets we have to use some kind of higher-orderlogic.

If one opts for periods as primitive entities, the simplest mathematicalmodeling will involve structures consisting of a set P of periods equippedwith a collection of natural relations on P . But in contrast to the point-basedapproach where the temporal precedence relation is the candidate for such arelation, we now have many options. For instance, since one is obviously stillinterested in temporal precedence, the relation ≺, with p ≺ q holding if theentire period p precedes the entire period q, is a natural candidate, but so isthe inclusion relation <, with p < q holding if p is a proper part of q. Andin fact, one widespread period-based modeling of time is that in structuresof the form P = (P,≺,<). But ≺ and < are not the only candidates. If weare interested in relations that are close to our common sense experience,then the relation of one period overlapping with another is quite relevantas well. And we are not confined to binary relations at all: we may needa unary predicate informing us whether a period is of zero duration (andhence, point-like), whereas there are also interesting ternary relation suchas the relation C holding of a triple p, q, r if p can be ‘chopped’ into thetwo pieces q and r. Of course, just like in the point-based case, one needsto impose conditions on period structures to make them useful as modelsof time. For instance, in a structure of the kind P = (P,≺,<) one willwant ≺ and < to be strict partial orders that are related by conditions like∀xyz (x < y ≺ z → x ≺ z) and others.

The reader may have realized how hard it is to gather one’s intuitionsand make a complete list of such conditions without taking resort to talkingabout points after all. The concept of a point in time has obviously beenvery useful in our thinking about time. Hence, even if periods are to be

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taken as the primitive entities of one’s ontology, it is at least interesting, ifnot a test for the viability of the proposal, to see whether one can constructpoint-based flows of time from period structures. Various ways have beenworked out for this purpose. Perhaps the simplest method is to take aspoints those periods that have zero duration — of course, this only works ifsuch entities are around and we have access to this information (for instance,through a zero-duration predicate as we mentioned above). But even if ourperiod structure does not have atomic periods, there are ways to extract apoint structure from it, for instance, by defining a point to be any maximalset of mutually overlapping pairs of periods. Finally, once there are ways toconstruct point structures from period structures and vice versa, the obviousquestion is to see how such constructions interact. This line of research hasbeen taken up with great mathematical sophistication, in a number of caseseven leading to interesting categorical dualities.

Interval-based temporal logic Just as in the case for point-based tem-poral logics, we may choose a class of period structures, design a formallanguage to talk about it, and study the resulting temporal logic.

For instance, suppose that we are working with intervals in point-basedflows of time, as described above. Taking the modal approach, we find our-selves in a multi-dimensional setting; that is, we want to evaluate formulasat pairs of points representing the beginning and the end point of the inter-val, respectively. Typical modal operators are 〈D〉 and ◦ with truth tablesgiven by

M, [s, t] 〈D〉ϕ if M, [u, v] ϕ for some t, u with s ≤ u ≤ v ≤ t,M, [s, t] ϕ ◦ ψ if M, [s, u] ϕ and M, [u, t] ψ for some u with s ≤ u ≤ t.

In words, 〈D〉ϕ holds at an interval if ϕ holds at some interval during it,while ϕ ◦ ψ holds at an interval if can be chopped into a ϕ- and a ψ-part.In period terms, one would say that v and C are the accessibility relationsof 〈D〉 and ◦, respectively.

For such modal systems, one may investigate meta-logical propertieslike completeness and decidability. The general picture here is that one hasa price to pay for the increase in expressivity: complete axiomatizationsare scarce and hard to find, and undecidability is the rule rather than theexception. On a technical level, the modal logic of time periods thus seemsto be more complex (and hence, more intriguing) than point logics over thesame flows of time, but the kinds of questions that are asked do not differmuch.

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Hence, let us finish this section mentioning some issues that are of specificinterest to period logics. To start with, period logics differ from point logicsin the sense that in many cases it is natural to correlate the interpretationof atomic propositions. A condition that one often encounters is that ofhomogeneity requiring that an atomic proposition holds at a period if andonly if it holds at each of its parts. It is obvious that such a conditiononly has intuitive appeal for the propositions corresponding to the eventcategories of states and activities. And even in the latter case, one mayraise objections to the ‘only if’ part of this condition: I can truthfully saythat I have been walking through town for hours when in fact, I have pauseda couple of times to take a coffee.

Now suppose that we are implementing this condition on some intervalstructure I(T ) induced by the flow of time T by demanding that for eachpropositional variable p and each point-based valuation π we have

(I(T ), π), [s, t] p iff π(u)(p) = 1 for all u with s ≤ u ≤ t.(6)

Observe that thus we have effectively reduced period predicates to pointpredicates. Such a reduction would have considerable computational ad-vantages, something that can easily be explained by taking a first orderperspective. It is obvious that the particular proposal (6) is rather naive:Zeno’s moving arrow will lead us into trouble. But perhaps there are moreinventive modellings in which formulas can be evaluated at periods, whilethe valuations remain point-based?

In any case, regardless of the technical advantages of reducing periodpredicates to point predicates, it is clear that there is a rather general philo-sophical issue at stake here, namely the problem of which kinds of predicatesapply to periods and points, respectively, and how these are correlated. Thisissue is in fact a matter of ongoing, and at times heated, debate.

6 Temporal Logic Now

As we mentioned before, temporal logic has become a vast and active re-search area with applications in many disciplines. In this section we willbriefly sketch some of these recent developments. Since not all of the workmentioned here is covered by the monographs mentioned at the end of thenext section, we provide references to the literature.

Richer ontological structures One common trend in temporal logic isto study logics of richer ontological structures since it is obvious that for se-

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rious real-world applications the kind of temporal logics that we have beendescribing until now are far too simple. For example, one shortcoming ofstandard temporal logics is that they only deal with qualitative timing prop-erties, whence they are inadequate for applications such as reasoning aboutreal-time behaviour of software. In order to overcome this deficiency, peoplehave designed logics for describing two-sorted structures consisting of a lin-ear flow of time connected with some metric domain. Such approaches canbe found both in the point-based and in the period-based paradigm, cf. [16]and [10], respectively. Another example of a multiple sorted ontology wehave already met in the semantics of Ockhamist branching time logic, wherebranches appeared as a second kind of entities, next to points. One mightvary on this ‘standard’ Ockhamist logic by admitting only some instead ofall branches, perhaps a collection satisfying some addition constraints [24].Applying this idea of using multiple sorted temporal ontologies to the dis-cussion of the previous section, one can envisage structures in which points,periods and events co-exist, linked by suitable relations [18]. One possibil-ity for such a link involves the notion of granularity: atomic objects mightsuddenly turn out to be divisible when approached at a different level. Thisobviously ties up with our way of classifying periods of time (months, weeks,days); modal logics for such layered structures are described in [15].

Temporal logic at work Turning temporal logics into actual workingsystems has created a number of interesting problems and challenges. Forinstance, one of the most fundamental contributions that Artificial Intelli-gence has made to the field of temporal logic, is that of identifying the frameproblem. This is the problem of formalizing the properties of an applicationarea that are unaffected by the performance of some action without explicitlysumming up all such properties. This problem appears to be independentof the particular formalism employed, and has to be faced by anyone wish-ing to give a formal account of reasoning about change [19]. The computerscience literature on modal logics of time has yielded an interesting perspec-tive on the modal truth relation (M, t ϕ) between a model M, whichis supposed to be finite, and a formula ϕ; in this perspective ϕ representssome property of a program and M some implementation of the program.For obvious reasons then, a considerable amount of effort has been devotedto finding fast model checking algorithms deciding whether a given formulaholds in a given finite model [21]. As a last example we mention the dynamicturn which research in the semantics of natural language has taken. In thisway of thinking, the meaning of a formula does not lie so much in its truth

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condition; linguistic expressions are rather like programs that update theinformation state of some agent. For instance, in Discourse RepresentationTheory [12] temporal expressions in natural language are used to extendand refine temporal representations of the discourse; these representationson their turn are syntactic items themselves that can be interpreted in stan-dard models.

Temporal logic in context There is an increasing tendency to studymodal formalisms not as isolated systems but in connection with otherbranches of logic, as in Correspondence Theory which relates modal logicto first and second order logic. For instance, the use of game-theoreticmethods has deepened our understanding of the relative expressive powerof modal logics of time: in particular, variants of Ehrenfeucht-Fraısse gameshave provided an interesting perspective on expressive completeness resultssuch as Theorem 4.1 [11, 22]. Recent approaches to decidability questionsconcerning modal and temporal logics use insights from algebraic logic andautomata theory. This has lead to the identification of a variety of decidablefragments of first order logic, each of which is obtained from atomic formulasusing all boolean connectives but allowing only a specific, guarded patternof quantification [1]. As a last example we mention the emergence of so-called hybrid languages which aim to boost the expressive power of modallanguages by adding some features from first order logic like ‘names’ (specialvariables that are to be true at a single state), over which quantification isallowed [9, 4].

7 Epilogue

What then, is temporal logic?In the narrowest sense temporal logic comprises the design and study of

specific systems for representing and reasoning about time, such as Prior’stense logic. These enterprises may have both an applied and a theoreticalside, the former consisting of designing a system (that is, making choicesin the fields of ontology, syntax and semantics), formalizing temporal phe-nomena in it, and then putting it to work (perhaps through implementingit). On the theoretical side, one aims at proving formal properties of thesystem, such as completeness or decidability.

On a slightly wider scale, temporal logicians may thus provide a supplyof general tools and techniques for answering questions pertaining to specificsystems. As an example we mention the method of filtration which is a quite

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general method of proving decidability of a temporal logic, and the canonicalmodel method which is very useful in proving completeness results.

A more ambitious aim for temporal logicians is to come up with frame-works for comparing and connecting different modellings of time. This aimcan be realized both at a technical and at a philosophical level. As an ex-ample of the first, think of the game-theoretic analysis of the expressivepower of modal languages, or of the duality between point and period-basedrepresentations of time, respectively. On a philosophical level, a thoroughclassification of event types and of the correlation between predicates per-taining to points and to periods, respectively, would be an extremely usefultool in any discussion on formal representations of temporal phenomena.

Since all of this is relevant for each of the disciplines where formal rea-soning about time is needed, Temporal Logic forms a prime example of thegrowing role of Logic as a source and channel of ideas and techniques appli-cable in related disciplines. Ultimately, one would hope that temporal logiccan provide a unifying perspective on our sometimes confusing thoughtsabout this highly puzzling thing we call time.

Suggested Further Reading In this chapter we have only scratched thesurface of Temporal Logic. The following monographs, each surveying partof the field of temporal logic, would form a good start for a bibliography.Concerning the philosophy of time I do not believe there is one standardreference, but Whitrow [23] is a very comprehensive study of the concept oftime, while Le Poidevin & MacBeath [13] brings together some seminalarticles on the subject. Øhstrøm & Hasle [17] gives a good treatmentof philosophical aspects of temporal logic from a historical perspective. InGoldblatt [8] the reader finds a concise and very accessible treatment ofthe most important modal logics of time; Gabbay et alii [5] is a moreextensive mathematical treatment. Manna & Pnueli [14] is a classic onapplications of temporal logic in computer science; Gabbay et alii [6] givesa good overview of the applications of temporal logic in artificial intelligence.There seems to be no monograph on the treatment in formal linguisticsof temporal aspects of natural language, but Steedman [20] surveys thefield well. Van Benthem [3] is a stimulating blend of much of the above.Finally, for an overview of recent developments in temporal logic the readeris referred to the proceedings of the first two conferences devoted solely totemporal logic, ICTL’94 [7] and ICTL’97 [2].

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Acknowledgements The research of the author has been made possibleby a fellowship of the Royal Netherlands Academy of Arts and Sciences.Personal thanks are due to Johan van Benthem for helpful comments on adraft version of this chapter, to Marco Aiello for a scrutinous reading of themanuscript, and to Lou Goble for extensive help in cutting the manuscriptto an acceptable size.

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[3] J. van Benthem. The Logic of Time. Kluwer, Dordrecht, second edition,1991.

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